Planes Of Square Order Research Articles

Partitioning the projective plane into two incidence‐rich parts

AbstractAn internal or friendly partition of a vertex set of a graph is a partition to two nonempty sets such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.

On the stability of Baer subplanes

A blocking set in a projective plane is a point set intersecting each line. The smallest blocking sets are lines. The second smallest minimal blocking sets are Baer subplanes (subplanes of order q). Our aim is to study the stability of Baer subplanes in PG(2,q). If we delete q+1−k points from a Baer subplane, then the resulting set has size q+k and (q+1−k)(q−q) 0-secants. If we have somewhat more 0-secants, then our main theorem says that this point set can be obtained from a Baer subplane or from a line by deleting somewhat more than q+1−k points and adding some points. The motivation for this theorem comes from planes of square order, but our main result is valid also for non-square orders. Hence in this case the point set contains a relatively large collinear subset.

Some blocking semiovals of homology type in planes of square order

A blocking semioval in a projective plane is a set of points which is both a semioval and a blocking set. In this paper, blocking semiovals in the Desaguesian projective plane $$\mathrm{PG}(2,s^2)$$ PG ( 2 , s 2 ) admitting an order $$s+1$$ s + 1 homology group are considered. The geometry of the point-orbits of such a group is studied. Using this geometry two new blocking semiovals are constructed in $$\mathrm{PG}(2,5^2)$$ PG ( 2 , 5 2 ) . Their automorphism group is also discussed.

Planes in which every quadrangle lies on a unique Baer subplane

Desarguesian projective planes of square order are characterized by the property that every quadrangle lies on a unique Baer subplane.

Polarities and unitals in the Coulter–Matthews planes

For a class of finite shift planes introduced by Coulter and Matthews, we give a set of representatives for the isomorphism types, determine all automorphisms and describe all polarities explicitly. The planes in question are the only known examples of finite shift planes that are not translation planes. Each non-desarguesian Coulter---Matthews plane admits precisely two conjugacy classes of orthogonal polarities. In addition, each Coulter---Matthews plane of square order admits exactly one conjugacy class of unitary polarities. We prove that most of the corresponding unitals are not classical.

The correlations of finite Desarguesian planes of square order defined by diagonal matrices

The present article represents the next step in our ongoing program of classifying the correlations of finite Desarguesian planes. We show that in PG ( 2 , q 2 n ) , the correlations defined by diagonal matrices, with companion automorphism ( q m ), where ( m , 2 n ) = 1 , have the following numbers of absolute points: q 2 n + q n + 2 - q n + 1 + 1 or q 2 n - q n + 1 + q n + 1 or ( q n + 1 ) 2 for n odd; q 2 n - q n + 2 + q n + 1 + 1 or q 2 n + q n + 1 - q n + 1 or ( q n - 1 ) 2 for n even. We also discuss the equivalence classes into which these correlations fall, as well as the configurations of their sets of absolute points.

Sets of type-(1, n) in biplanes

A set of type- (m,n) S is a set of points of a design with the property that each block of the design meets either m points or n points of S. If m=1, S gives rise to a subdesign of the design. The parameters of sets of type-(1, n) in finite projective planes were characterised by G. Tallini and M. Tallini Scafati with more generalised order condition. It follows from their result that, a set of type-(1, n) exists only in the planes of square orders and it gives rise to either a Baer subplane or a unital of a finite projective plane of square order. In this paper, we characterise the parameters of sets of type-(1, n) in biplanes of more extended order condition than prime power. It follows from the results that a set of type-(1, n) in a biplane is either a Baer subdesign, a Hermitian subdesign or a subdesign with certain types of parameters. In addition, some examples of sets of type-(1, n) are given in known biplanes.

On small dense arcs in Galois planes of square order

In the Galois projective plane of square order q, we show the existence of small dense ( k,4)-arcs whose points lie on two conics for q odd, and on two hyperovals for q even. We provide an explicit construction of (4 q −4,2) -arcs for q even, and we also show that they are complete as far as q⩽1024.

Ternary dual codes of the planes of order nine

We determine the minimum weight of the ternary dual codes of each of the four projective planes of order 9, and of the seven affine planes of order 9. The proof includes a construction of a word of small, sometimes minimal, weight in the dual code of any plane of square order containing a Baer subplane.

The non-existence of certain large minimal blocking sets

Recently, Blokhuis and Metsch [6] proved that a Desarguesian projective plane of square order q ≥ 25 contains no minimal blocking -set. Since for q = 4 the existence is proved in [2], the problem is open when q ∈ {9, 16}, see also [4]. In this paper we prove the theorem for q = 9. Since the techniques are of combinational type, based on incidence properties, the result holds also in non-Desarguesian cases. The length of the proof shows the difficulty to treat this subject in a general finite projective plane.

Some maximal arcs in Hall planes

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of square order, including the Desarguesian ones, but not the Hall planes. We construct a family of maximal arcs in the Hall planes inherited from the Thas maximal arcs in the Desarguesian planes. In particular, maximal arcs are shown to exist in all Hall planes of even order.

A characterisation of thas maximal arcs in translation planes of square order

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of order q2. We characterise these as being exactly those (non-trivial) maximal arcs that are stabilised by an homology of order q− 1.

Generalizing a non-existence theorem in finite projective planes

We show that the sets of absolute points of correlations of Desarguesian projective planes do not possess the so-called Pasch (O'Nan) configuration. Previously, this has been known of unitary polarities of planes of square order.

Abelian Projective Planes of Square Order

We prove that, under certain conditions, multipliers of an abelian projective plane of square order have odd order modulo v *, where v * is the exponent of the underlying Singer group. As a consequence, we are able to establish the non-existence of an infinite number of abelian projective planes of square order.

On a class of derived translation planes of square order

A class of planes π of order q 2 derived from the class of planes π [ Discrete Math. 66:175–190 (1987)] of order q 2, where q = p r , p is a prime, p ⩾ 7, p ≢ ± 1 (mod 10), and r is an odd natural number, is studied. All these planes π , except when q = 7, 13, share the property that the translation complement divides the set of distinguished points into five orbits of lengths 1, 1, ( q − 1)/2, ( q − 1)/2, and q 2 − q. In the case q = 7 the translation complement divides the set of distinguished points into two orbits of lengths 8 and 42. In the case of q = 13, the translation complement partitions the set of distinguished points into three orbits of lengths 2, 12, and 156. These orbit structures make these planes π distinct from the translation planes of the class of planes π. Further, the class of planes π are identified as the third class of the three classes of planes of Cohen and Ganley [ Quart. J. Math. Oxford 35:101–113 (1984)]. Thus the third class of planes of Cohen and Ganley may be obtained from the class of planes π by derivation.

On a class of translation planes of square order

A class of translation planes of order q 2, where q = p r , p is a prime, p ⩾7, p ≠± 1 (mod 10) and r is an odd natural number is constructed and the translation complements of these planes are determined. A property shared by all these planes is that the translation complement fixes a distinguished point and divides the remaining distinguished points into two orbits of length q and q 2 − q. The order of the translation complement is rq( q − 1) 2 except for q = 7 and q = 13. The translation complements of these exceptional cases are also briefly studied. The class of planes considered in this paper are distinct from the classes of translation planes of S.D. Cohen and M.J. Ganley [Quart. J. Math. Oxford, 35 (1984) 101–113].

A class of translation planes of square order

A class of translation planes of order p2r, where r is an odd natural number and p is a prime, p ≥ 7, p ≢ ± (mod 10) is constructed. A salient feature shared by all these planes is that one ideal point is fixed by the translation complement and the remaining ideal points are divided into at least two orbits, one of which is of length pr.

On the maximum sizes of certain (k, n)-arcs in finite projective geometries

In this paper the values of m(6)2,9, m(5)2,8 and m(5)2,9 have been determined as well as improved bounds on other arcs, and in particular a general construction for certain large arcs in planes of square order has been given. Table 5 summarizes the known values of m(n)2, q for 2 ≤ n ≤ q and q ≤ 9.

Characterizations of the Generalized Hughes Planes

Let be a projective plane and a subplane of . If l is a line of , we let denote the group of all elations in that have as axis and leave Q invariant. In [12, p. 921], Ostrom asked for a description of all finite planes that have a Baer subplane with the property that for all lines l of . Here denotes the order of G. Both the desarguesian planes of square order and the generalized Hughes planes have this property (Hughes [10], Ostrom [14], Dembowski [6]). One of the aims of this paper is to show that these are the only planes having such a Baer subplane.

Rank 3 affine planes of square order

Rank 3 affine planes of square order